Let G, H be graphs. A set I of vertices of G is called an H-isolating set of G if \(G-N[I]\) contains no copies of H, where N[I] is the closed neighborhood of I. The cardinality of a minimum H-isolating set in G is called the H-isolation number of G and is denoted by \(\iota (G,H)\) . Zhang and Wu (Discrete Appl Math 357:99–111, 2024) proved that, given a connected graph H, for any graph G with exactly s components, \(\iota (G,H)\le \gamma (H)\frac{m(G)+s}{m(H)+1}\) , where \(\gamma (H)\) is the domination number of H, and m(G) and m(H) are the respective sizes of G and H. Then, they conjectured that, given a graph H with \(\gamma (H)=1\) , for any connected graph \(G\ne H\) , \(\iota (G,H)\le \frac{m(G)+1}{m(H)+2}\) . Borg (Discrete Appl Math 371:247–253, 2025) proved the conjecture. In this paper, we prove that, given a connected graph H with \(\gamma (H)\ge 2\) , for any graph G, \(\iota (G,H)\le \gamma (H)\frac{m(G)}{m(H)+\gamma (H)+1}\) with equality if and only if \(H=P_4\) and every component of G is \(P_4\) . The result is a natural extension of Borg’s one and an improvement of Zhang and Wu’s one for \(\gamma (H)\ge 2\) . Several problems are posed.
